two simple facts about well-founded relations
The following are two simple facts about well-founded relation R on X:
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1.
For each x∈X, xR̸x. (See the entry R-minimal element.)
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2.
The requirement for symmetry is absent, i.e., for each x,y∈X, either xRy or yRx, but not both.
Justifications for these two facts are simple. For 1, consider the subclass {x}. Then {x} has an R-minimal element, which can only be x itself. For 2, consider {x,y}. It has an R-minimal element, which is either x or y, not both.
Fact 1 is provided here for easy reference. Keeping these two facts in mind is helpful when dealing with (proving) basic theorems about the relation.
Title | two simple facts about well-founded relations |
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Canonical name | TwoSimpleFactsAboutWellfoundedRelations |
Date of creation | 2013-03-22 18:25:43 |
Last modified on | 2013-03-22 18:25:43 |
Owner | yesitis (13730) |
Last modified by | yesitis (13730) |
Numerical id | 10 |
Author | yesitis (13730) |
Entry type | Feature |
Classification | msc 03E20 |